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series RC circuit

Kevin Weddle

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Anybody ever notice that if you apply, let's say a triangle, to a series RC circuit, the rate of change of voltage on one side of the capacitor is different from the rate of change of voltage on other side because of reduced amplitude? Even phase shift can't correct this. This means the capacitor impedance might vary depending on the circuit values.

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Hi Kevin,
A triangle wave consists of many harmonics. The resistor passes them all at the same level. The reactance of the parallel capacitor is a lower impedance at high frequencies than the resistor so it passes the higher frequencies more than the lower frequencies.

An RC circuit with the same time constants that is a series resistor then a capacitor to ground (de-emphasis) will be a perfect correction for the pre-emphasis effect of the parallel network.

FM and TV stations use pre-emphasis (treble boost) during audio transmission and FM radios and TVs use de-emphasis (treble and hiss cut) to return the sound back to normal, but with the hiss from the radio system reduced.

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If a triangle produces stronger harmonics than a sinewave, the measuring device would not be accurate.

The AC mode of a multimeter is made to measure the sinewave of the mains frequency. It is inaccurate at higher frequencies or with waveforms having higher frequency harmonics.
A 'scope or AC millivoltmeter can accurately measure higher frequencies or waveforms with harmonics.

A sine-wave is a single frequency without any harmonics.
A triangle wave is a fundamental sine-wave frequency plus many harmonic sine-waves.
A square-wave also has a fundamental sine-wave plus many stronger harmonic sine-waves.
A triangle wave and a square wave are both symmetrical so contain only odd-numbered harmonics.

A highpass coupling capacitor or a lowpass filter capacitor passes or attenuates the high frequency harmonics differently than the lower frequency fundamental frequency so the shape and sound of the waveform is different.
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I know any waveform can be duplicated with sinewaves of various phase freqeuncy, and amplitude. Maybe harmonics is the wrong language here. I'm looking at a sinewave which approaches zero rate of change. It does this gradually. So maybe it does not have harmonics, but it does have a lot of change in voltage that is of a different rate of change than other parts of the sinewave. Maybe we'd say it has a very wide bandwidth. A triangle though, the rate of change is constant except for a corner, which has a low rate of change but little change in voltage.

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I think also that this is where theory leaves electronics. An oscillator fabricated from electronic circuits will produce harmonics, but a perfect sine on paper has none.

The very low distortion sine-wave oscillator I made has harmonics at such a low level that is is nearly impossible to measure them. The strongest harmonic is the 3rd and it is at 1/50,000th of the output level. Most of the even-numbered harmonics are absent.

I took a CD4018 Walking-Ring Counter and made a stepped sine-wave having 10 steps using 4 resistors. Then I filtered it with an 8th-order Butterworth lowpass filter made from two switched capacitor lowpass filter ICs. Then I filtered it some more with a 2nd-order Butterworth Sallen and Key lowpass filter made with a dual TL072 opamp. If I used a better opamp then the very small amount of harmonics would be even less.

Opamp manufacturers measure the distortion from their products and measure extremely low levels of the harmonics. The same with audio amplifier manufacturers.
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I think Kevin's question is expressed in terms of time, and should thus be replied to in terms of time. I reckon the trouble understanding reactive components (like capacitors) for most people lies in the difference between understanding their behaviour in terms of frequency and understanding them in terms of time. The two approaches are so far removed from each other that you might think we are talking about different components altogether.

In the time domain, a capacitor's voltage changes at a rate proportional to the current flowing through it. Thus:

Firstly, the capacitor's voltage cannot change instantly. It can only gradually approach some value, as it charges or discharges. The higher the time constant (R x C) of an RC circuit, the slower the rate of voltage change across the capacitor. If an input signal changes slowly enough, the capacitor in an RC circuit is able to charge and discharge quickly enough to keep up with input changes. If the input changes too quickly though, the capacitor cannot charge or discharge fast enough to follow the input. This inability of the capacitor's voltage to swing quickly enough results in its voltage being an attenuation of the input.

Secondly, intuitively, it can be seen that if the input changes significantly faster than the capacitor can follow, fluctuations of voltage across the capacitor will be negligible compared to fluctuations in input voltage. Thus it is the resistor that is dominant in determining the current through the network, and so the current is roughly proportional to the input voltage. This means that the rate of change of the capacitor voltage is proportional to the instantaneous input voltage. This is the cause of waveform distortion (not harmonic distorion).

So, in a low pass RC circuit, the output is the capacitor's voltage, whose rate of change is (nearly) proportional to the instantaneous input voltage, and the circuit is said to integrate. Conversely, with the resistor and capacitor swapped to form a high-pass filter, the ouput is the resistor's voltage, and the circuit differentiates, so that the ouput at any instant is (nearly) proportional to the rate of change of input. Read that again.

The upshot of all this is that not only does the RC network attenuate, but it also distorts, by either integrating or differentiating. So, for a low-pass RC circuit, a square wave input (whose period is well below the R x C time constant of the circuit) will appear heavily attenuated at the output. It will also be distorted into a triangle wave, because the alternating high and low input voltages are causing the capacitor to charge and discharge (nearly) linearly - otherwise known as integration.

For that same circuit, fed with a nice sinusoid of period significantly lower than R x C, the sinusoid is integrated. The integration of sin(x) is another sinusoid, phase shifted by -90 degrees (in other words, an upside down cosine, or -cos(x)). A high-pass RC filter would phase shift by +90 degrees, to yield a cosine. Try plotting a sinusoid, and then the rate of change of that sinusoid, and you'll see this effect clearly - two sinuoids out of phase with each other by 90 degrees.

An interesting experiment to demonstrate this is to connect an oscilloscope in X-Y mode to the circuit - channel 1 to the input, and channel 2 to the output. This yields a wonderful ellipse, or nearly a circle if you choose your input signal frequency and channels' vertical scales properly.

In summary, when feeding a simple RC network with a sinusoid, you get another phase-shifted sinusoid across the capacitor. With non-sinusoidal input waveforms, the capacitor voltage is always a distortion of the input. The amount of distortion depends upon how far the capacitor's rate of charge/discharge is exceeded by rate of change of input. Periodic signals whose periods are significantly smaller than the time constant R x C will appear in attenuated and integrated form across the capacitor, and in attenuated and differentiated form across the resistor.

Understanding an RC network in these terms (the time domain) permits an understanding of how timing circuits (like the 555 IC) work, but is not really appropriate for understanding filter applications. Those are best described in the frequency domain, because the input waveform is rarely some nice square, triangular or sinusoidal form.

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